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The uploaded document contains a series of mathematical problems mostly centered around arithmetic and geometric sequences. Let's address each question one by one:

1. **Question**: The first few terms of a sequence are given as 4.9, 14.9, 24.9, ... The task is to identify the common difference and the formula for the nth term of the arithmetic sequence.
   - **Solution**: The common difference (d) can be found by subtracting the first term from the second term: \(d = 14.9 - 4.9 = 10\). The nth term of the arithmetic sequence can be expressed as \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term. Thus, \(a_n = 4.9 + (n-1) \times 10\).

2. **Question**: Determine the nth term of the sequence where the second term is 17.1, the fourth term is 17.3, and the terms form an arithmetic sequence.
   - **Solution**: The common difference is calculated as \(d = (17.3 - 17.1) / 2 = 0.1\). The nth term formula becomes \(a_n = 17 + 0.1n\), setting the first term \(a_1 = 17\) (because \(a_2 = 17.1\)).

3. **Question**: If the 7th term of a sequence is 20, and the 15th term is 3, find the nth term.
   - **Solution**: Using the arithmetic sequence formula \(a_n = a_1 + (n-1)d\), with the given \(a_7 = 20\) and \(a_{15} = 3\), we find \(d = (3 - 20) / (15 - 7) = -17/8 = -2.125\). The first term \(a_1\) can be determined by substituting back into the formula for any term.

4. **Question**: The sequence describes the number of chairs in rows, increasing by three each row: 14 chairs, 17 chairs, etc. The task is to find the total number of chairs in the 20th row.
   - **Solution**: This forms an arithmetic sequence where \(a_1 = 14\), \(d = 3\). For the 20th row, \(a_{20} = 14 + 19 \times 3 = 71\).

5. **Question**: Calculate the sum of the first ten terms of the sequence \(-5, -8, -11, -14, \ldots\).
   - **Solution**: This is an arithmetic sequence where \(a_1 = -5\), \(d = -3\). The sum of the first 10 terms \(S_{10}\) can be calculated using the sum formula \(S_n = \frac{n}{2} \times (2a_1 + (n-1)d)\).

6. **Question**: Given \(a_6 = 17\) and \(a_{10} = 33\) of an arithmetic sequence, find \(S_{20}\).
   - **Solution**: First find \(d\) and \(a_1\) using the given terms, then use \(S_n = \frac{n}{2} \times (2a_1 + (n-1)d)\).

7. **Question**: For a sequence starting with 168 and decreasing, the task is to calculate \(a_{12}\).
   - **Solution**: Given \(S_{12} = 168\), use the sum formula in reverse to find \(a_1\) and \(d\), and subsequently find \(a_{12}\).

These are the solutions for the first seven questions. Would you like to proceed with the remaining questions or need detailed explanations for any specific question among these?

The image contains multiple questions related to arithmetic and geometric sequences, including finding terms, sums, and rules for sequences.

This page explores how to solve problems related to arithmetic and geometric sequences, including calculating the nth term, sums, and understanding sequence properties. Suitable for high school students (Grades 9-12).

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